.file "sinhf.s" // Copyright (c) 2000 - 2005, Intel Corporation // All rights reserved. // // Contributed 2000 by the Intel Numerics Group, Intel Corporation // // Redistribution and use in source and binary forms, with or without // modification, are permitted provided that the following conditions are // met: // // * Redistributions of source code must retain the above copyright // notice, this list of conditions and the following disclaimer. // // * Redistributions in binary form must reproduce the above copyright // notice, this list of conditions and the following disclaimer in the // documentation and/or other materials provided with the distribution. // // * The name of Intel Corporation may not be used to endorse or promote // products derived from this software without specific prior written // permission. // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR // A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL INTEL OR ITS // CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, // EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, // PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR // PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY // OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY OR TORT (INCLUDING // NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS // SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. // // Intel Corporation is the author of this code, and requests that all // problem reports or change requests be submitted to it directly at // http://www.intel.com/software/products/opensource/libraries/num.htm. // History //********************************************************************* // 02/02/00 Initial version // 04/04/00 Unwind support added // 08/15/00 Bundle added after call to __libm_error_support to properly // set [the previously overwritten] GR_Parameter_RESULT. // 10/12/00 Update to set denormal operand and underflow flags // 01/22/01 Fixed to set inexact flag for small args. // 05/02/01 Reworked to improve speed of all paths // 05/20/02 Cleaned up namespace and sf0 syntax // 11/20/02 Improved algorithm based on expf // 03/31/05 Reformatted delimiters between data tables // // API //********************************************************************* // float sinhf(float) // // Overview of operation //********************************************************************* // Case 1: 0 < |x| < 2^-60 // Result = x, computed by x+sgn(x)*x^2) to handle flags and rounding // // Case 2: 2^-60 < |x| < 0.25 // Evaluate sinh(x) by a 9th order polynomial // Care is take for the order of multiplication; and A2 is not exactly 1/5!, // A3 is not exactly 1/7!, etc. // sinh(x) = x + (A1*x^3 + A2*x^5 + A3*x^7 + A4*x^9) // // Case 3: 0.25 < |x| < 89.41598 // Algorithm is based on the identity sinh(x) = ( exp(x) - exp(-x) ) / 2. // The algorithm for exp is described as below. There are a number of // economies from evaluating both exp(x) and exp(-x). Although we // are evaluating both quantities, only where the quantities diverge do we // duplicate the computations. The basic algorithm for exp(x) is described // below. // // Take the input x. w is "how many log2/128 in x?" // w = x * 64/log2 // NJ = int(w) // x = NJ*log2/64 + R // NJ = 64*n + j // x = n*log2 + (log2/64)*j + R // // So, exp(x) = 2^n * 2^(j/64)* exp(R) // // T = 2^n * 2^(j/64) // Construct 2^n // Get 2^(j/64) table // actually all the entries of 2^(j/64) table are stored in DP and // with exponent bits set to 0 -> multiplication on 2^n can be // performed by doing logical "or" operation with bits presenting 2^n // exp(R) = 1 + (exp(R) - 1) // P = exp(R) - 1 approximated by Taylor series of 3rd degree // P = A3*R^3 + A2*R^2 + R, A3 = 1/6, A2 = 1/2 // // The final result is reconstructed as follows // exp(x) = T + T*P // Special values //********************************************************************* // sinhf(+0) = +0 // sinhf(-0) = -0 // sinhf(+qnan) = +qnan // sinhf(-qnan) = -qnan // sinhf(+snan) = +qnan // sinhf(-snan) = -qnan // sinhf(-inf) = -inf // sinhf(+inf) = +inf // Overflow and Underflow //********************************************************************* // sinhf(x) = largest single normal when // x = 89.41598 = 0x42b2d4fc // // Underflow is handled as described in case 1 above // Registers used //********************************************************************* // Floating Point registers used: // f8 input, output // f6,f7, f9 -> f15, f32 -> f45 // General registers used: // r2, r3, r16 -> r38 // Predicate registers used: // p6 -> p15 // Assembly macros //********************************************************************* // integer registers used // scratch rNJ = r2 rNJ_neg = r3 rJ_neg = r16 rN_neg = r17 rSignexp_x = r18 rExp_x = r18 rExp_mask = r19 rExp_bias = r20 rAd1 = r21 rAd2 = r22 rJ = r23 rN = r24 rTblAddr = r25 rA3 = r26 rExpHalf = r27 rLn2Div64 = r28 rGt_ln = r29 r17ones_m1 = r29 rRightShifter = r30 rJ_mask = r30 r64DivLn2 = r31 rN_mask = r31 // stacked GR_SAVE_PFS = r32 GR_SAVE_B0 = r33 GR_SAVE_GP = r34 GR_Parameter_X = r35 GR_Parameter_Y = r36 GR_Parameter_RESULT = r37 GR_Parameter_TAG = r38 // floating point registers used FR_X = f10 FR_Y = f1 FR_RESULT = f8 // scratch fRightShifter = f6 f64DivLn2 = f7 fNormX = f9 fNint = f10 fN = f11 fR = f12 fLn2Div64 = f13 fA2 = f14 fA3 = f15 // stacked fP = f32 fT = f33 fMIN_SGL_OFLOW_ARG = f34 fMAX_SGL_NORM_ARG = f35 fRSqr = f36 fA1 = f37 fA21 = f37 fA4 = f38 fA43 = f38 fA4321 = f38 fX4 = f39 fTmp = f39 fGt_pln = f39 fWre_urm_f8 = f40 fXsq = f40 fP_neg = f41 fX3 = f41 fT_neg = f42 fExp = f43 fExp_neg = f44 fAbsX = f45 RODATA .align 16 LOCAL_OBJECT_START(_sinhf_table) data4 0x42b2d4fd // Smallest single arg to overflow single result data4 0x42b2d4fc // Largest single arg to give normal single result data4 0x00000000 // pad data4 0x00000000 // pad // // 2^(j/64) table, j goes from 0 to 63 data8 0x0000000000000000 // 2^(0/64) data8 0x00002C9A3E778061 // 2^(1/64) data8 0x000059B0D3158574 // 2^(2/64) data8 0x0000874518759BC8 // 2^(3/64) data8 0x0000B5586CF9890F // 2^(4/64) data8 0x0000E3EC32D3D1A2 // 2^(5/64) data8 0x00011301D0125B51 // 2^(6/64) data8 0x0001429AAEA92DE0 // 2^(7/64) data8 0x000172B83C7D517B // 2^(8/64) data8 0x0001A35BEB6FCB75 // 2^(9/64) data8 0x0001D4873168B9AA // 2^(10/64) data8 0x0002063B88628CD6 // 2^(11/64) data8 0x0002387A6E756238 // 2^(12/64) data8 0x00026B4565E27CDD // 2^(13/64) data8 0x00029E9DF51FDEE1 // 2^(14/64) data8 0x0002D285A6E4030B // 2^(15/64) data8 0x000306FE0A31B715 // 2^(16/64) data8 0x00033C08B26416FF // 2^(17/64) data8 0x000371A7373AA9CB // 2^(18/64) data8 0x0003A7DB34E59FF7 // 2^(19/64) data8 0x0003DEA64C123422 // 2^(20/64) data8 0x0004160A21F72E2A // 2^(21/64) data8 0x00044E086061892D // 2^(22/64) data8 0x000486A2B5C13CD0 // 2^(23/64) data8 0x0004BFDAD5362A27 // 2^(24/64) data8 0x0004F9B2769D2CA7 // 2^(25/64) data8 0x0005342B569D4F82 // 2^(26/64) data8 0x00056F4736B527DA // 2^(27/64) data8 0x0005AB07DD485429 // 2^(28/64) data8 0x0005E76F15AD2148 // 2^(29/64) data8 0x0006247EB03A5585 // 2^(30/64) data8 0x0006623882552225 // 2^(31/64) data8 0x0006A09E667F3BCD // 2^(32/64) data8 0x0006DFB23C651A2F // 2^(33/64) data8 0x00071F75E8EC5F74 // 2^(34/64) data8 0x00075FEB564267C9 // 2^(35/64) data8 0x0007A11473EB0187 // 2^(36/64) data8 0x0007E2F336CF4E62 // 2^(37/64) data8 0x00082589994CCE13 // 2^(38/64) data8 0x000868D99B4492ED // 2^(39/64) data8 0x0008ACE5422AA0DB // 2^(40/64) data8 0x0008F1AE99157736 // 2^(41/64) data8 0x00093737B0CDC5E5 // 2^(42/64) data8 0x00097D829FDE4E50 // 2^(43/64) data8 0x0009C49182A3F090 // 2^(44/64) data8 0x000A0C667B5DE565 // 2^(45/64) data8 0x000A5503B23E255D // 2^(46/64) data8 0x000A9E6B5579FDBF // 2^(47/64) data8 0x000AE89F995AD3AD // 2^(48/64) data8 0x000B33A2B84F15FB // 2^(49/64) data8 0x000B7F76F2FB5E47 // 2^(50/64) data8 0x000BCC1E904BC1D2 // 2^(51/64) data8 0x000C199BDD85529C // 2^(52/64) data8 0x000C67F12E57D14B // 2^(53/64) data8 0x000CB720DCEF9069 // 2^(54/64) data8 0x000D072D4A07897C // 2^(55/64) data8 0x000D5818DCFBA487 // 2^(56/64) data8 0x000DA9E603DB3285 // 2^(57/64) data8 0x000DFC97337B9B5F // 2^(58/64) data8 0x000E502EE78B3FF6 // 2^(59/64) data8 0x000EA4AFA2A490DA // 2^(60/64) data8 0x000EFA1BEE615A27 // 2^(61/64) data8 0x000F50765B6E4540 // 2^(62/64) data8 0x000FA7C1819E90D8 // 2^(63/64) LOCAL_OBJECT_END(_sinhf_table) LOCAL_OBJECT_START(sinh_p_table) data8 0x3ec749d84bc96d7d // A4 data8 0x3f2a0168d09557cf // A3 data8 0x3f811111326ed15a // A2 data8 0x3fc55555552ed1e2 // A1 LOCAL_OBJECT_END(sinh_p_table) .section .text GLOBAL_IEEE754_ENTRY(sinhf) { .mlx getf.exp rSignexp_x = f8 // Must recompute if x unorm movl r64DivLn2 = 0x40571547652B82FE // 64/ln(2) } { .mlx addl rTblAddr = @ltoff(_sinhf_table),gp movl rRightShifter = 0x43E8000000000000 // DP Right Shifter } ;; { .mfi // point to the beginning of the table ld8 rTblAddr = [rTblAddr] fclass.m p6, p0 = f8, 0x0b // Test for x=unorm addl rA3 = 0x3E2AA, r0 // high bits of 1.0/6.0 rounded to SP } { .mfi nop.m 0 fnorm.s1 fNormX = f8 // normalized x addl rExpHalf = 0xFFFE, r0 // exponent of 1/2 } ;; { .mfi setf.d f64DivLn2 = r64DivLn2 // load 64/ln(2) to FP reg fclass.m p15, p0 = f8, 0x1e3 // test for NaT,NaN,Inf nop.i 0 } { .mlx // load Right Shifter to FP reg setf.d fRightShifter = rRightShifter movl rLn2Div64 = 0x3F862E42FEFA39EF // DP ln(2)/64 in GR } ;; { .mfi mov rExp_mask = 0x1ffff fcmp.eq.s1 p13, p0 = f0, f8 // test for x = 0.0 shl rA3 = rA3, 12 // 0x3E2AA000, approx to 1.0/6.0 in SP } { .mfb nop.m 0 nop.f 0 (p6) br.cond.spnt SINH_UNORM // Branch if x=unorm } ;; SINH_COMMON: { .mfi setf.exp fA2 = rExpHalf // load A2 to FP reg nop.f 0 mov rExp_bias = 0xffff } { .mfb setf.d fLn2Div64 = rLn2Div64 // load ln(2)/64 to FP reg (p15) fma.s.s0 f8 = f8, f1, f0 // result if x = NaT,NaN,Inf (p15) br.ret.spnt b0 // exit here if x = NaT,NaN,Inf } ;; { .mfi // min overflow and max normal threshold ldfps fMIN_SGL_OFLOW_ARG, fMAX_SGL_NORM_ARG = [rTblAddr], 8 nop.f 0 and rExp_x = rExp_mask, rSignexp_x // Biased exponent of x } { .mfb setf.s fA3 = rA3 // load A3 to FP reg nop.f 0 (p13) br.ret.spnt b0 // exit here if x=0.0, return x } ;; { .mfi sub rExp_x = rExp_x, rExp_bias // True exponent of x fmerge.s fAbsX = f0, fNormX // Form |x| nop.i 0 } ;; { .mfi nop.m 0 // x*(64/ln(2)) + Right Shifter fma.s1 fNint = fNormX, f64DivLn2, fRightShifter add rTblAddr = 8, rTblAddr } { .mfb cmp.gt p7, p0 = -2, rExp_x // Test |x| < 2^(-2) fma.s1 fXsq = fNormX, fNormX, f0 // x*x for small path (p7) br.cond.spnt SINH_SMALL // Branch if 0 < |x| < 2^-2 } ;; { .mfi nop.m 0 // check for overflow fcmp.ge.s1 p12, p13 = fAbsX, fMIN_SGL_OFLOW_ARG mov rJ_mask = 0x3f // 6-bit mask for J } ;; { .mfb nop.m 0 fms.s1 fN = fNint, f1, fRightShifter // n in FP register // branch out if overflow (p12) br.cond.spnt SINH_CERTAIN_OVERFLOW } ;; { .mfi getf.sig rNJ = fNint // bits of n, j // check for possible overflow fcmp.gt.s1 p13, p0 = fAbsX, fMAX_SGL_NORM_ARG nop.i 0 } ;; { .mfi addl rN = 0xFFBF - 63, rNJ // biased and shifted n-1,j fnma.s1 fR = fLn2Div64, fN, fNormX // R = x - N*ln(2)/64 and rJ = rJ_mask, rNJ // bits of j } { .mfi sub rNJ_neg = r0, rNJ // bits of n, j for -x nop.f 0 andcm rN_mask = -1, rJ_mask // 0xff...fc0 to mask N } ;; { .mfi shladd rJ = rJ, 3, rTblAddr // address in the 2^(j/64) table nop.f 0 and rN = rN_mask, rN // biased, shifted n-1 } { .mfi addl rN_neg = 0xFFBF - 63, rNJ_neg // -x biased, shifted n-1,j nop.f 0 and rJ_neg = rJ_mask, rNJ_neg // bits of j for -x } ;; { .mfi ld8 rJ = [rJ] // Table value nop.f 0 shl rN = rN, 46 // 2^(n-1) bits in DP format } { .mfi shladd rJ_neg = rJ_neg, 3, rTblAddr // addr in 2^(j/64) table -x nop.f 0 and rN_neg = rN_mask, rN_neg // biased, shifted n-1 for -x } ;; { .mfi ld8 rJ_neg = [rJ_neg] // Table value for -x nop.f 0 shl rN_neg = rN_neg, 46 // 2^(n-1) bits in DP format for -x } ;; { .mfi or rN = rN, rJ // bits of 2^n * 2^(j/64) in DP format nop.f 0 nop.i 0 } ;; { .mmf setf.d fT = rN // 2^(n-1) * 2^(j/64) or rN_neg = rN_neg, rJ_neg // -x bits of 2^n * 2^(j/64) in DP fma.s1 fRSqr = fR, fR, f0 // R^2 } ;; { .mfi setf.d fT_neg = rN_neg // 2^(n-1) * 2^(j/64) for -x fma.s1 fP = fA3, fR, fA2 // A3*R + A2 nop.i 0 } { .mfi nop.m 0 fnma.s1 fP_neg = fA3, fR, fA2 // A3*R + A2 for -x nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fP = fP, fRSqr, fR // P = (A3*R + A2)*R^2 + R nop.i 0 } { .mfi nop.m 0 fms.s1 fP_neg = fP_neg, fRSqr, fR // P = (A3*R + A2)*R^2 + R, -x nop.i 0 } ;; { .mfi nop.m 0 fmpy.s0 fTmp = fLn2Div64, fLn2Div64 // Force inexact nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fExp = fP, fT, fT // exp(x)/2 nop.i 0 } { .mfb nop.m 0 fma.s1 fExp_neg = fP_neg, fT_neg, fT_neg // exp(-x)/2 // branch out if possible overflow result (p13) br.cond.spnt SINH_POSSIBLE_OVERFLOW } ;; { .mfb nop.m 0 // final result in the absence of overflow fms.s.s0 f8 = fExp, f1, fExp_neg // result = (exp(x)-exp(-x))/2 // exit here in the absence of overflow br.ret.sptk b0 // Exit main path, 0.25 <= |x| < 89.41598 } ;; // Here if 0 < |x| < 0.25. Evaluate 9th order polynomial. SINH_SMALL: { .mfi add rAd1 = 0x200, rTblAddr fcmp.lt.s1 p7, p8 = fNormX, f0 // Test sign of x cmp.gt p6, p0 = -60, rExp_x // Test |x| < 2^(-60) } { .mfi add rAd2 = 0x210, rTblAddr nop.f 0 nop.i 0 } ;; { .mmb ldfpd fA4, fA3 = [rAd1] ldfpd fA2, fA1 = [rAd2] (p6) br.cond.spnt SINH_VERY_SMALL // Branch if |x| < 2^(-60) } ;; { .mfi nop.m 0 fma.s1 fX3 = fXsq, fNormX, f0 nop.i 0 } { .mfi nop.m 0 fma.s1 fX4 = fXsq, fXsq, f0 nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fA43 = fXsq, fA4, fA3 nop.i 0 } { .mfi nop.m 0 fma.s1 fA21 = fXsq, fA2, fA1 nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fA4321 = fX4, fA43, fA21 nop.i 0 } ;; // Dummy multiply to generate inexact { .mfi nop.m 0 fmpy.s0 fTmp = fA4, fA4 nop.i 0 } { .mfb nop.m 0 fma.s.s0 f8 = fA4321, fX3, fNormX br.ret.sptk b0 // Exit if 2^-60 < |x| < 0.25 } ;; SINH_VERY_SMALL: // Here if 0 < |x| < 2^-60 // Compute result by x + sgn(x)*x^2 to get properly rounded result .pred.rel "mutex",p7,p8 { .mfi nop.m 0 (p7) fnma.s.s0 f8 = fNormX, fNormX, fNormX // If x<0 result ~ x-x^2 nop.i 0 } { .mfb nop.m 0 (p8) fma.s.s0 f8 = fNormX, fNormX, fNormX // If x>0 result ~ x+x^2 br.ret.sptk b0 // Exit if |x| < 2^-60 } ;; SINH_POSSIBLE_OVERFLOW: // Here if fMAX_SGL_NORM_ARG < x < fMIN_SGL_OFLOW_ARG // This cannot happen if input is a single, only if input higher precision. // Overflow is a possibility, not a certainty. // Recompute result using status field 2 with user's rounding mode, // and wre set. If result is larger than largest single, then we have // overflow { .mfi mov rGt_ln = 0x1007f // Exponent for largest single + 1 ulp fsetc.s2 0x7F,0x42 // Get user's round mode, set wre nop.i 0 } ;; { .mfi setf.exp fGt_pln = rGt_ln // Create largest single + 1 ulp fma.s.s2 fWre_urm_f8 = fP, fT, fT // Result with wre set nop.i 0 } ;; { .mfi nop.m 0 fsetc.s2 0x7F,0x40 // Turn off wre in sf2 nop.i 0 } ;; { .mfi nop.m 0 fcmp.ge.s1 p6, p0 = fWre_urm_f8, fGt_pln // Test for overflow nop.i 0 } ;; { .mfb nop.m 0 nop.f 0 (p6) br.cond.spnt SINH_CERTAIN_OVERFLOW // Branch if overflow } ;; { .mfb nop.m 0 fma.s.s0 f8 = fP, fT, fT br.ret.sptk b0 // Exit if really no overflow } ;; // here if overflow SINH_CERTAIN_OVERFLOW: { .mfi addl r17ones_m1 = 0x1FFFE, r0 fcmp.lt.s1 p6, p7 = fNormX, f0 // Test for x < 0 nop.i 0 } ;; { .mmf alloc r32 = ar.pfs, 0, 3, 4, 0 // get some registers setf.exp fTmp = r17ones_m1 fmerge.s FR_X = f8,f8 } ;; { .mfi mov GR_Parameter_TAG = 128 (p6) fnma.s.s0 FR_RESULT = fTmp, fTmp, f0 // Set I,O and -INF result nop.i 0 } { .mfb nop.m 0 (p7) fma.s.s0 FR_RESULT = fTmp, fTmp, f0 // Set I,O and +INF result br.cond.sptk __libm_error_region } ;; // Here if x unorm SINH_UNORM: { .mfb getf.exp rSignexp_x = fNormX // Must recompute if x unorm fcmp.eq.s0 p6, p0 = f8, f0 // Set D flag br.cond.sptk SINH_COMMON // Return to main path } ;; GLOBAL_IEEE754_END(sinhf) LOCAL_LIBM_ENTRY(__libm_error_region) .prologue { .mfi add GR_Parameter_Y=-32,sp // Parameter 2 value nop.f 0 .save ar.pfs,GR_SAVE_PFS mov GR_SAVE_PFS=ar.pfs // Save ar.pfs } { .mfi .fframe 64 add sp=-64,sp // Create new stack nop.f 0 mov GR_SAVE_GP=gp // Save gp };; { .mmi stfs [GR_Parameter_Y] = FR_Y,16 // Store Parameter 2 on stack add GR_Parameter_X = 16,sp // Parameter 1 address .save b0, GR_SAVE_B0 mov GR_SAVE_B0=b0 // Save b0 };; .body { .mfi stfs [GR_Parameter_X] = FR_X // Store Parameter 1 on stack nop.f 0 add GR_Parameter_RESULT = 0,GR_Parameter_Y // Parameter 3 address } { .mib stfs [GR_Parameter_Y] = FR_RESULT // Store Parameter 3 on stack add GR_Parameter_Y = -16,GR_Parameter_Y br.call.sptk b0=__libm_error_support# // Call error handling function };; { .mmi add GR_Parameter_RESULT = 48,sp nop.m 0 nop.i 0 };; { .mmi ldfs f8 = [GR_Parameter_RESULT] // Get return result off stack .restore sp add sp = 64,sp // Restore stack pointer mov b0 = GR_SAVE_B0 // Restore return address };; { .mib mov gp = GR_SAVE_GP // Restore gp mov ar.pfs = GR_SAVE_PFS // Restore ar.pfs br.ret.sptk b0 // Return };; LOCAL_LIBM_END(__libm_error_region) .type __libm_error_support#,@function .global __libm_error_support#